Einstein Field Equation: Understanding Curvature and Geodesic Equations

• TimeRip496
In summary, the EFE is an equation that allows us to calculate the curvature caused by mass and energy, and the Schwarzschild metric is one particular solution to it. While any coordinates can be used in solving the EFE, spherical coordinates are often preferred due to the simplicity of the equations. The geodesic equation is used to determine the path of an object in a gravitational field by integrating it.
TimeRip496
Is EFE an equation to help us find the curvature cause by mass? Like the schwarzchild metric? In addition, is it a must to use polar coordinate for EFE to work since it contains dr? Can we used the ordinary Euclidean coordinate in minkoswki space for EFE? If we can't is it because there is no radius(dr) component in it? Cause curvature results in excess radius, right?

Lastly, geodesic equation is the one that allows us to determine the path of an object in gravitational field by integrating it, right?

TimeRip496 said:
Is EFE an equation to help us find the curvature cause by mass? Like the schwarzchild metric?
Yes, the EFE allows us to calculate the curvature caused by the presence of mass (and energy). The Schwarzschild metric is one particular solution to the EFE, the one that you get if you apply the EFE to the vacuum outside of a spherical mass.

In addition, is it a must to use polar coordinate for EFE to work since it contains dr? Can we used the ordinary Euclidean coordinate in minkoswki space for EFE? If we can't is it because there is no radius(dr) component in it?
You can use any coordinates you wish in solving the EFE - it holds true in all coordinate systems. However, in any given problem some coordinates will be much easier less hard to use than others. We use spherical instead of Minkowski coordinates to derive the Scwarzschild solution for the same reason that we use spherical instead of Euclidean coordinates in the Newtonian solution for planetary orbits - for these particular problems the equations are a lot simpler written in spherical coordinates.If you haven't already tried working through http://preposterousuniverse.com/grnotes/grtinypdf.pdf , give it a try. This is about as simple of an introduction as you will find (some will say that it's too simple); if you can't follow the math there, that's a pretty good hint that you'll need more math background before you're ready to go up against the EFE.

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1. What is the Einstein field equation?

The Einstein field equation is a set of 10 equations that make up the core of Einstein's theory of general relativity. It describes the relationship between the curvature of space-time and the distribution of matter and energy within it.

2. How does the Einstein field equation relate to gravity?

The Einstein field equation explains gravity as the curvature of space-time caused by the presence of mass and energy. In this theory, objects with mass create a curvature in space-time, and other objects then follow this curvature, resulting in the force we know as gravity.

3. What is the significance of understanding curvature in space-time?

Understanding curvature in space-time is crucial for understanding the behavior of objects in the universe. It allows us to describe how gravity works and make predictions about the movement of celestial bodies, such as planets and stars.

4. How are geodesic equations related to the Einstein field equation?

The geodesic equations are a set of equations that describe the path that an object will take in space-time. These equations are derived from the Einstein field equation and are used to determine the trajectory of objects in a curved space-time.

5. Is there any experimental evidence for the accuracy of the Einstein field equation?

Yes, there is a significant amount of experimental evidence that supports the accuracy of the Einstein field equation. Some examples include the precession of Mercury's orbit, the bending of light by massive objects, and the observations of gravitational waves. These all provide strong evidence for the validity of Einstein's theory of general relativity and the Einstein field equation.

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